On the asymptotics of whispering gallery waves for crystal groups with even order axes of symmetry

The generic approach for representing, in anisotrpic case, the wave field of surface modes by means of caustic expansions (that is modified ray series), is applied to specific types of symmetry (i.e., anisotropy) of elastic media, where the existence of even axes of symmetry is supposed. Under assumption that the boundary surface S is the crystal´s plane of symmetry, which is orthogonal to the even axis of symmetry, the surface modes (in zeroth approximations of asymptotics) turn out to be polarized along the normal n⃗ to S, i.e., co-directed with the axis of symmetry. Two other quasi-shear and quasi-longitudinal waves exhibit as inhomogeneous evanescent with depth waves, expressed in the form of ray series with complex eikonals and with their amplitudes as correction terms to the amplitude of a surface wave. Namely the presence of these waves in asymptotic solution provide the boundary conditions to be fulfilled. Taking this into account, the resulting formulas for amplutudes and eikonals of these waves are deduced. And also, for crystals of tetragonal and cubic syngonies, the cases of multiple characterics are considered, where phase velocities of shear and guasi-shear waves coincide at some points on the surface S.